Articles récents

Reflective tomography is an efficient method for optical imaging in the visible and near infrared ranges. It computes empirical reconstructions based on algorithms from X-ray tomography. This subject introduces mathematical gaps to be filled, about the meaning of the reconstructions, and about their artifacts. To tackle these questions, we study more generally the filtered backprojection on projections outside the range of the Radon transform. We consider generic projections that can involve any kind of physical and geometric parameters. We claim that the backprojection contains partially the geometry of the original scene. More precisely, we compare the singularities of the backprojection with the singularities of a representation of the scene. This comparison of wavefront sets, inspired by studies of the artifacts in X-ray tomography, is based on microlocal analysis. It gives a precise meaning to the well-reconstructed geometry, describes the invisible parts, and the artifacts. We illustrate the heuristic and the analysis principle on canonical cases that belong to various fields: shape from silhouettes, constructible tomography, cloaking, reconstruction from cartoon images, imaging of occluded lambertian objects. Numerical results show the relevance of the heuristic and its analysis. In a word, this study provides a mathematical framework that covers the solver of reflective tomography, and exhibits an imaging method whose range of application is wide.

The intracranial pressure (ICP) is an important factor in the proper functioning of the brain. This pressure is needed to be constantly regulated, since an abnormal elevation can be quite dangerous. In this article, we develop some numerical tools to better understand the regulation of this pressure. In particular, as it is impossible to measure the ICP in a non-invasive way, these numerical tools can allow to estimate values of the ICP. In addition, we propose to compute the dynamics of the cerebrospinal fluid (CSF), taking into account the connected environment of the skull and the arterio-venous flows. A computational fluid dynamics model in two dimensions is developed for the cerebrospinal fluid system, with Windkessel type boundary conditions. This model shows that the dynamics can impact the distribution of the CSF in the different compartments of the cerebrospinal system.